Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

8.3: Putting it all together

( \newcommand{\kernel}{\mathrm{null}\,}\)

In summary, we have u(ρ,ϕ)=A02+n=1ρn(Ancosnϕ+Bnsinnϕ).

The one remaining boundary condition can now be used to determine the coefficients An and Bn, U(c,ϕ)=A02+n=1cn(Ancosnϕ+Bnsinnϕ)={100if 0<ϕ<π0if π<ϕ<2π.
We find A0=1ππ0100dϕ=100,cnAn=1ππ0100cosnϕdϕ=100nπsin(nϕ)|π0=0,cnBn=1ππ0100sinnϕdϕ=100nπcos(nϕ)|π0={200/(nπ)if n is odd0if n is even.
In summary u(ρ,ϕ)=50+200πn odd(ρc)nsinnϕn.
We clearly see the dependence of u on the pure number r/c, rather than ρ. A three dimensional plot of the temperature is given in Fig. 8.3.1.

Temp0-100.png
Figure 8.3.1: The temperature (8.3.2)

This page titled 8.3: Putting it all together is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?